Planar Josephson Hall Effect in Topological Josephson Junctions

PHYSICAL REVIEW B 103, 054508 (2021)

Planar Josephson Hall effect in topological Josephson junctions

Oleksii Maistrenko ,1,* Benedikt Scharf ,2 Dirk Manske,1 and Ewelina M. Hankiewicz2,† 1Max-Planck-Institut für Festkörperforschung, D-70569 Stuttgart, Germany 2Institute for Theoretical Physics and Astrophysics and Würzburg-Dresden Cluster of Excellence ct.qmat, University of Würzburg, Am Hubland, 97074 Würzburg, Germany

(Received 11 March 2020; revised 1 February 2021; accepted 2 February 2021; published 17 February 2021)

Josephson junctions based on three-dimensional topological insulators offer intriguing possibilities to realize unconventional p-wave pairing and Majorana modes. Here, we provide a detailed study of the effect of a uniform magnetization in the normal region: We show how the interplay between the spin-momentum locking of the topological insulator and an in-plane magnetization parallel to the direction of phase bias leads to an asymmetry of the Andreev spectrum with respect to transverse momenta. If sufﬁciently large, this asymmetry induces a transition from a regime of gapless, counterpropagating Majorana modes to a regime with unprotected modes that are unidirectional at small transverse momenta. Intriguingly, the magnetization-induced asymmetry of the Andreev spectrum also gives rise to a Josephson Hall effect, that is, the appearance of a transverse Josephson current. The amplitude and current phase relation of the Josephson Hall current are studied in detail. In particular, we show how magnetic control and gating of the normal region can enable sizable Josephson Hall currents compared to the longitudinal Josephson current. Finally, we also propose in-plane magnetic ﬁelds as an alternative to the magnetization in the normal region and discuss how the planar Josephson Hall effect could be observed in experiments.

DOI: 10.1103/PhysRevB.103.054508

I. INTRODUCTION Topological Josephson junctions are particularly intriguing if they are based on 3D TIs, as depicted in Fig. 1(a): The helical spin structure of the surface states of three- Because of the 2D nature of the surface, the system sup- dimensional topological insulators (3D TIs) offers intriguing ports modes that propagate along the direction parallel to the possibilities of tailoring the surface-state properties by vari- superconductor/normal TI interface, that is, the y direction ous proximity effects. A conventional s-wave superconductor in Fig. 1(a). Due to the protected zero-energy crossing oc- can, for example, be used to proximity-induce superconduc- curring at zero transverse momentum and phase difference tivity in the TI surface. The interplay between the helical φ = π,aπ junction exhibits two counterpropagating gapless spin-momentum locking of the TI surface state and the super- states, so-called nonchiral Majorana modes [1][seeFig.1(b) conducting pairing then mediates an effective pairing between bottom]. electrons at the Fermi level. This effective pairing features a Besides proximity-induced superconductivity, one can also mixture of singlet s-wave and triplet p-wave pair correlations envision other proximity effects whose interplay with the [1–3] and turns the TI surface into a topological superconduc- spin texture of the TI surface state leads to novel phenom- tor [2,4–9] with Majorana zero modes [1] and odd-frequency ena: In nonsuperconducting setups, for example, the interplay pairing [10]. between the helical surface states and proximity-induced mag- In this context, Josephson junctions based on 3D TIs or netism provides a versatile platform for studying fundamental on their two-dimensional (2D) counterparts have been studied effects and spintronic applications [4,32,33]. Ferromagnetic extensively for potential signatures of topological supercon- tunnel junctions based on 3D TIs [34–38], in particular, ductivity, both theoretically [1,3,11–24] and experimentally show some promise for potential spintronic devices [39]. The [25–29]. These so-called topological Josephson junctions ex- combination of 3D TIs with both proximity-induced super- hibit a ground-state fermion parity that is 4π-periodic in the conductivity and magnetism can prove even more interesting superconducting phase difference φ and Andreev bound states [11,40–42], however, and could point to novel possibilities for (ABS) with a protected zero-energy crossing [30,31]. superconducting spintronics [43,44]. Motivated by this prospect [45] as well as by phenomena found in nonsuperconducting TI tunneling junctions, such as *Corresponding author: [email protected] the tunneling planar Hall effect [38], we study 3D TI-based †Corresponding author: [email protected] Josephson junctions with a ferromagnetic tunneling barrier Published by the American Physical Society under the terms of the [see Fig. 1(c)]. In contrast to previous studies on this system Creative Commons Attribution 4.0 International license. Further [11,40,41], we focus not only on the longitudinal response distribution of this work must maintain attribution to the author(s) but also on the transverse response to an applied phase bias. and the published article’s title, journal citation, and DOI. Open We ﬁnd that especially the conﬁguration with an in-plane access publication funded by the Max Planck Society. magnetization parallel to the direction of the phase bias

2469-9950/2021/103(5)/054508(16) 054508-1 Published by the American Physical Society OLEKSII MAISTRENKO et al. PHYSICAL REVIEW B 103, 054508 (2021)

II. MODEL A. Hamiltonian and unitary transformation In our model, we consider a Josephson junction based on the 2D surface state of a 3D TI, as depicted in Fig. 1(c), where the pairing in the superconducting (S) regions is induced from a nearby s-wave superconductor. The ferromagnetic (F) region is subject to an exchange splitting/Zeeman term proximity induced from a nearby ferromagnet [48]. If one is only interested in an in-plane Zeeman term, an alternative way to realize such a Zeeman term is by applying an in-plane magnetic field as discussed in Sec. VII below. The surface state lies in the xy plane, with the direction of the superconducting phase bias denoted as the x direction. We take the system to be infinite in both the x and y directions. Here, we study the regime FIG. 1. (a) Scheme of a Josephson junction based on a 3D TI: where the Fermi level is situated inside the bulk gap and s-wave superconductors (S) on top of the TI proximity-induced where only surface states exist. Moreover, we assume that pairing into the TI surface state. The two proximity-induced super- the surface considered is far enough away from the opposite conducting regions are separated by a normal region. (b) Top view surface so that there is no overlap between their states. Then, and low-energy Andreev spectrum of a short topological π junction the Josephson junction based on a single surface is described for transverse momenta py close to py = 0: The two low-energy by the Bogoliubov-de Gennes (BdG) Hamiltonian ABS correspond to counterpropagating nonchiral Majorana modes with opposite group velocities. No net Josephson Hall current flows 0 Hˆ = [vF (σx pˆy − σy pˆx ) − μ]τz + (V0τz − M · σ)h(x) in the y direction. (c) Ferromagnetic Josephson junction based on BdG a 3D TI: Same as (a) but with a magnetic region separating the + (x)[τx cos (x) − τy sin (x)] (1) superconducting regions. In this setup, the Zeeman field/exchange splitting M is proximity induced by a ferromagnet (F). (d) Same † † T with the basis order ˆ = (ψˆ↑, ψˆ↓, ψˆ , −ψˆ ) .InEq.(1), as (a) but for a ferromagnetic Josephson junction with large M ↓ ↑ pˆl (with l = x, y) denote momentum operators and σl and parallel to the direction of the phase bias: The Andreev spectrum is τ = , , asymmetric and has been tilted in such a way that the two low-energy l (with l x y z) Pauli matrices in spin and particle-hole space, respectively. Moreover, σ = (σx,σy,σz ) and unit ma- modes are unidirectional for small py. Note that these ABS are no Majorana modes protected against backscattering because there are trices are not written explicitly in Eq. (1). In this paper, we study two models for a Josephson junc- additional zero-energy states for py close to the Fermi momentum (not shown). The asymmetry in the Andreev spectrum gives rise to a tionwithaFregionofwidthd: (a) a model with a δ-like finite Josephson Hall current flowing in the y direction. F region described by h(x) = dδ(x) and (x) = and (b) a model with a finite F region where h(x) = (d/2 −|x|) and (x) = (|x|−d/2). In both cases, the phase conven- tion is (x) = φ (x), where φ is the superconducting phase exhibits striking features: Such a magnetization leads to an difference between the two S regions. The superconducting asymmetric Andreev spectrum for a fixed finite transverse pairing amplitude with strength 0 is proximity induced momentum. If sufficiently large, this asymmetry even induces from the s-wave superconductors deposited on the TI surface. a transition from the regime of counterpropagating, nonchiral The density of states of these superconductors is typically Majorana modes to a regime with unprotected unidirectional much larger than that of the TI surface states, e.g., we can modes at small transverse momenta [compare Fig. 1(b) bot- assume Nb superconductors used in experiments. Therefore, tom and Fig. 1(d) bottom]. Most importantly, even a small the currents flowing in the TI surface do not significantly magnetization-induced asymmetry in the Andreev spectrum affect the superconducting phases and we can use constant φ causes a transverse Josephson Hall current [see Fig. 1(d) top]. and within each superconducting lead. In other words the In contrast to other Josephson Hall effects [46,47], the effect resistivity of the junction region is much larger than that of found here arises from an in-plane magnetization, which is the leads; this justifies our approximation [49,50] commonly why we call it the planar Josephson Hall effect. The planar used in mesoscopic systems. We note that the superconductors Josephson Hall effect is the superconducting analog to the from which superconductivity is induced in the TI surface tunneling planar Hall effect found in nonsuperconducting TI states are not explicitly included in our model (1). However, tunneling junctions [38]. in the real system they are important to make this assumption. Below, we will discuss the origin of the Josephson Hall The Fermi velocity of the surface state is vF , and V0 the current, its properties, and how it could be experimentally potential in the F region, which can also be viewed as de- verified. The paper is organized as follows: After introducing scribing the difference between the chemical potentials in the the effective model used to describe the Josephson junction S and F regions, μ and μF = μ − V0. The Zeeman term due in Sec. II, we study its ABS in Sec. III. In Secs. IV and V, to the proximity-induced ferromagnetic exchange splitting is the procedure to compute the different Josephson currents is described by the effective magnetization M = (Mx, My, Mz ) presented. These currents are then discussed in Sec. VI.A [48]. Note that the direction of M is set by the magnetization brief summary concludes the paper in Sec. VII. in the ferromagnet.

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For our calculations, it is more convenient to introduce locking will have important consequences for the discussion the unitary√ rotation transformation in spin space U = (1 − below. iσz )/ 2 and bring the Dirac Hamiltonian (1) into the form III. ANDREEV BOUND STATES HˆBdG = [σ · (vF pˆ − τzM h(x)) − μ + V0h(x)]τz A. General equations + (x)[τx cos (x) − τy sin (x)] (2) In order to understand the Josephson currents and the emer- with pˆ = (ˆpx, pˆy, 0). Because of the rotated spin axes used gence of a Josephson Hall current, it is instructive to first in Eq. (2) M is a rotated effective magnetization, which is look at the ABS of Eq. (3), that is, bound states decaying for related to the components of the real magnetization M induced |x|→∞and hence with energies |E| <. We focus on the in the F region via M = (−M , M , M ). In addition, for the y x z ABS of a junction with finite F region and refer to Appendix Dirac equation this Zeeman term has the same form as a vector A for the Andreev spectrum of the δ model, where relatively potential. From now on, we use the Hamiltonian (2) because compact, analytical solutions are possible in certain limiting it proves more convenient mathematically. cases. The eigenenergies of the ABS and their corresponding eigenstates can be determined from the ansatz B. General form of the solutions ⎧ ⎪A ψ (S) x + A ψ (S) x , x < − d ˆ = ⎪ 1 e,−sμ ( ) 2 h,sμ ( ) To solve HBdG (r) E (r) and obtain the eigenspectrum ⎨ 2 (F ) d of Eq. (2), we first make use of translational invariance along ψ x = Dξαψ (x), |x| < ( ) ⎪ ξ=e/h,α=± ξα 2 (7) the y direction, [HˆBdG, pˆy] = 0. Although, on a macroscopic ⎩⎪ B ψ (S) (x) + B ψ (S) (x), d < x scale y d the phase may depend on the transverse coor- 1 e,sμ 2 h,−sμ 2 dinate, this should not affect the local structure of Andreev for a junction with a finite F region and sμ = sgn(μ). Now, levels calculated below.√ Hence, we proceed with the ansatz the coefficients A1, A2, De±, Dh±, B1, B2 have to be calculated = ipyyψ / (r) e (x) W , which reasonably simplifies the an- from the boundary conditions at the S/F interfaces, p alytical treatment of the system. Here y is the momentum + − + − quantum number, ψ(x) is a spinor in Nambu space, and W ψ(0 ) = ψ(0 ),ψ(d ) = ψ(d ). (8) is a unit width of the system in the y direction. Even if one The boundary conditions (8) lead to systems of linear equa- considers a large finite-size system in the y direction, these tions for the coefficients A1 to B2. By requiring a nontrivial solutions should describe states away from the boundaries. solution of this system of linear equations, that is, by re- = Here and in the remainder of this paper, we seth ¯ 1. The quiring its determinant to vanish, we find the ABS energies ψ eigenenergies and (x) can then be obtained from the 1D E = E(φ, p ). BdG equation y

HˆBdG(py )ψ(x) = Eψ(x), (3) B. Andreev spectrum of a ferromagnetic Josephson junction where HˆBdG(py )isgivenbyEq.(2) with the operatorp ˆy This procedure enables us to compute the Andreev spec- replaced by the quantum number p . trum of a finite barrier, examples of which are shown in Fig. 2 y = The energy-momentum relation in the S regions is given by for a short junction with an F region of length d 330 nm, √ |μ| = μ ± 2/ 2 − 2 = 2 − 2 , and different configurations of M. For these param- q± ( ) vF py with E . We find the eters, there are two ABS with energies E±(φ, py )atagiven following solutions in the S leads: momentum py, where the subscript ± denotes which state lies φ, φ, 1 higher (lower) in energy, that is, E+( py ) E−( py ). We 1 uξ α ψ (S) = ⊗ α ξ + i qξ x, can compare the φ and py dependence of these ABS with ξα (x) √ −i(x) vF ( q ipy ) e (4) 2 e vξ μ + ξ the case of no magnetization, that is, M = 0 (not shown): For M = 0, the Andreev spectrum E±(φ, p ) exhibits a zero- ξ =± y where 1 corresponds to particlelike and holelike solu- energy crossing protected by fermion parity at odd integer tions and α =±1 selects the direction of motion. Here, φ = π = multiples of and py 0, as also discussed in Appendix C. This protected zero-energy crossing is accompanied by 1 ξ 1 ξ uξ = 1 + , vξ = 1 − . (5) two gapless, nonchiral Majorana modes that counterpropagate 2 E 2 E along the y direction and are localized mostly in the normal In the F region, the electron and hole states are given by region [1,3]. If we include a finite M, its effects on the ABS are the −ξ / +α ei( My vF kξ )x − ξ ψ (F ) = √ E Mz following: ξα (x) α + − ξ 2E (E − ξMz ) vF kξ i(vF py Mx ) (i) A component My (not shown) shifts the entire Andreev (6) spectrum as a function of φ, that is, E±(φ, py ) → E±(φ + = μ ± − 2 − ∓ 2 − 2 , = / with vF ke/h ( E V0 ) (vF py Mx ) Mz and 2Zy py ), where Zy Myd vF , but leaves the spectrum other- E = μ + ξE − V0. The Zeeman term Mz opens a symmetric wise unchanged [11]. In particular, the protected zero-energy gap in the spectrum, while in-plane magnetization Mx shifts crossing for py = 0 and the nonchiral Majorana modes are the position of the Dirac cone and introduces an asymmetry now shifted to φ = (2n + 1)π − 2Zy, where n ∈ Z is an inte- in the barrier states. For a given py mode this changes the ger. Indeed, the My component can be absorbed into the phase effective energy gap in the barrier, making the Andreev reflec- difference by performing a gauge transformation of the BdG tion process angle dependent. This result of spin-momentum Hamiltonian (see Appendix A).

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[42]. It is important to note that the unidirectional ABS close to py = 0 are, however, not protected against backscattering: As can be seen in Fig. 2(d), these states are accompanied by other zero-energy states with py close to the Fermi momentum and with opposite group velocities. The results presented above show that although the Zee- man term enters the equations in the form of a vector potential [see Eq. (2)], its effect is not limited to the semiclassical phase factor typical for a spin degenerate electron system. The found asymmetry of the Andreev spectrum emerges from the interplay between the spin-orbit coupling of the TI and Mx which plays the role of the magnetic tunneling barrier. We discuss it in more detail in Appendix B with an effective low-energy model. In Fig. 2, we also compare the numerically obtained ABS with the analytical expressions one can derive for the ABS of a model with a δ-like F region in the Andreev approximation, as discussed in Appendix A. For short junctions and momenta close to py = 0, these analytical expressions provide an excel- lent description of the ABS. In particular, these expressions also capture the asymmetry and ‘tilting’ of the Andreev spec- FIG. 2. Andreev bound state spectra for different combinations trum induced by Mx. In addition, we show in Appendix D2 = = . = of M and V0: (a),(c) M Mez, V0 1 5 meV and (b),(d) M Mex, that the same effect is also present for parameters beyond the = = V0 0. Here, el denotes a unit vector in the l direction with l Andreev approximation, i.e., for μ ∼ . x, y, z. In all panels, M = 0.2meV,d = 330 nm, μ = 2meV,vF = 5 × 105 m/s, and = 100 μeV. The solid lines depict the spectra given in Eq. (A8) for the δ barrier in the Andreev approximation. The C. Spin structure of Andreev bound states discrete data points depict the numerically computed ABS spectra Figure 3 shows the spatial dependence of the quasiparticle as obtained for the finite F region without any approximations to 2 density |ψ(x)| =σ0(x) ofthetwoABSforφ = 0.9π and Eq. (2). different momenta py if Mz = 0 [Figs. 3(a)–3(c)] and if Mx = 0 [Figs. 3(d)–3(f)]. As can be discerned from Figs. 3(a)–3(c), (ii) Finite components Mx and Mz, shown in Figs. 2(b) Mz = 0 leads to ABS that are increasingly localized at the S/F and 2(d) and Figs. 2(a) and 2(c), respectively, also do not interfaces as py or Mz are increased. One can understand this remove this zero-energy crossing for py = 0 and φ = (2n + behavior by recalling that a magnetization component in the 1)π − 2Zy. This crossing remains protected by the fermion z direction acts as a mass term that increasingly isolates the parity and cannot be removed by a finite Mx or Mz [30,31] left and right S regions. If the two S regions are completely (see also Appendix C). The main effect of a finite out-of-plane isolated from each other, that is, for Mz →∞, each S region magnetization Mz in the F region is to detach the ABS from the separately corresponds to a topological superconductor that continuum states with |E| >[see Fig. 2(a) and Appendix hosts one chiral Majorana mode at its boundary [1]. Hence, D1], consistent with the results found in Refs. [40,41]. the results in Figs. 3(a)–3(c) can be interpreted as the in- (iii) Intriguingly, we find that a finite Mx = 0 introduces termediate regime between M = 0 with nonchiral Majorana an asymmetry in the Andreev spectrum at finite py as shown modes that are completely delocalized inside the F region and in Figs. 2(c) and 2(d): It does no longer satisfy E±(φ, py ) = Mz →∞with one chiral Majorana mode at each of the S/F 2 2 −E∓(φ, py ), but only the weaker condition E±(φ, py ) = interfaces. Comparing |ψ(x)| for finite Mz with |ψ(x)| for a 2 −E∓(φ,−py ), dictated by the particle-hole symmetry of the finite Mx of the same strength, we find that |ψ(x)| is not as BdG formalism. In particular, Fig. 2(d), which shows the localized at the S/F interfaces for Mx, but rather constant in py dependence of the Andreev spectrum, illustrates that the the whole F region, as shown in Figs. 3(d)–3(f). asymmetry E±(φ, py ) =−E∓(φ, py ) manifests itself in a ‘tilt- We also depict the expectation values of the spin densities † † ing’ of the spectrum. If Mx is large enough, it can even ψ (x)σxψ(x) =σx (x) and ψ (x)σyψ(x) =σy(x) in Fig. 3. lead to a situation where the group velocities in y direction, The spin densities σx (x) and σy(x) are related to the charge vg ∝ ∂E±(φ, py )/∂ py, for ABS in the vicinity of py = 0 and currents in the x and y directions, respectively (see Sec. IV φ ≈ π − 2Zy have the same sign. Such a situation is shown below). By comparing right and left columns of Fig. 3,we in Fig. 2(d). In this regime, the ABS change from nonchiral, find that for in-plane magnetization σy(x) is delocalized counterpropagating Majorana modes to modes propagating within the F region. This is in contrast to the out-of-plane in the same direction for small py.Atsmallpy, the disper- case, where the σy(x) spin density and the wave functions sion of these ABS is reminiscent of the unidirectional modes are peaked near the S/F interfaces. Another important obser- found in noncentrosymmetric superconductors [51–53]orin vation is that for finite Mx the spin polarization amplitudes Rashba sandwiches [54]. An energy spectrum asymmetric in of the two Andreev levels are no longer equal. Together with the transverse momentum py can also appear at a single F/S the asymmetry of the Andreev spectrum for Mx = 0 discussed interface due to broken rotational symmetry by the Mx term above, a finite σy(x) such as in Fig. 3 gives rise to a finite net

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operator † ρˆ (r) = e ψˆσ (r)ψˆσ (r)(9) σ

1 τ σ or equivalently by the matrix 2 e z 0 in the Nambu basis with e denoting the electron charge. The time evolution of the density operator is given by the equation of motion ∂ρ/∂ˆ t = i[HˆBdG, ρˆ (r, t )]. After using the fermionic commutation relations for ﬁeld operators, this equation of motion can be written in the form of the continuity equation ∂ρˆ + ∇ˆj(r) = Sˆ(r). (10) ∂t Here, the quasiparticle part of the current density is proportional to the spin operator, analogous to the nonsuperconducting case for Dirac materials [38] 1 ˆj(r) = ev ˆ †(r)τ σˆ (r). (11) 2 F 0 The source term corresponding to the conversion of quasipar- ticles to Cooper pairs in the S leads is given by

† Sˆ(r) = (x)ˆ (r)[τx sin (x) + τy cos (x)]ˆ (r). (12) The expectation values of these one-body operators can be expressed as traces of the Green’s function which will be derived in the next section.

V. GREEN’S FUNCTION ANALYSIS A. General expressions and analytical continuation In this section, we briefly describe the procedure for con- structing the Green’s function of the junction Hamiltonian (2). We follow the McMillan approach [55] and derive it from the wave-function solutions of the system. Thus, we choose the energies |E| > ||, where Eq. (4) describes propagating states, and solve the scattering problem ⎧ (S) ⎪ψ (S) + ψ < d , σ ⎪ n (x) rnn (py ) (x) x FIG. 3. Expectation values i (x) obtained from the ABS wave ⎨ n< n 2 = = . = = (F ) d functions: (a)–(c) M Mez, V0 1 5 meV, (d)–(f) M Mex, V0 ψ = s (p )ψ (x) |x| < , n> (x) n>,< nn y n 2 (13) 0. Here, e denotes a unit vector in l direction with l = x, y, z.Inall ⎪ l ⎩ (S) d φ = . π = . = μ = = t (p )ψ (x) x > , panels, 0 9 , M 0 2meV,d 330 nm, 2meV,vF n> nn y n 2 5 × 105 m/s, and = 100 μeV. Light blue designates the normal = α,ξ ∈{ } (F) region. where the multi-index n ( ) n> corresponds to an incident state from the left with fixed py. Using the boundary conditions defined in Eq. (8)[orinEq.(A1)foraδ barrier], we find the reflection and transmission coefficients rnn (py ) and Josephson Hall current, even for small Mx. The emergence of tnn (py ) correspondingly. Analogously, we can obtain states ψ this Josephson Hall current will be discussed next. n< (x) corresponding to processes when there is a quasiparticle incident from the right part of the junction. These solutions determine the continuum spectrum of the system, located above the superconducting gap. Furthermore, we employ the ˆ T IV. CURRENT OPERATORS AND CONTINUITY same procedure for the transposed Hamiltonian HBdG to find EQUATIONS the conjugate states Having found ABS with a peculiar behavior for M = 0, T x Hˆ ψñ(x) = Eψñ(x), (14) we next study whether this gives characteristic signatures in BdG observable quantities, such as for example the Josephson cur- where the transpose operation acts on the Pauli matrices rent. In order to derive current density operators, we consider (Nambu space) and on the coordinate space (by replacing pˆ the continuity equation for the charge density defined by the with −pˆ). Afterward, we can write the Green’s function for a

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fixed py as an outer product of these solutions Eq. (17) and a term originating from Sˆ(x)intheS regions, has a constant value and is independent of the position x.For C ψ (x)ψ˜ (x ), x > x , n>,n< nn n n ˆ GR (x, x , E ) = (15) the transverse current, there is no contribution due to S(x). py C ψ (x)ψ˜ (x ), x < x , n<,n> nn n n Finally, we remark that the current densities computed from Eqs. (15) and (17) are the current densities for a situation where the position-independent coefficients Cnn should be where all states have equilibrium occupations without any = determined from the boundary condition at x x , external constraints. Therefore, Eqs. (15) and (17) describe the i current densities without conservation of the fermion parity. GR (x + 0+, x) − GR (x − 0+, x) = τ σ . (16) py py z x vF B. Induced superconducting pairing Having determined the Green’s function of the system in this way, we can express a given single-particle operator in Before discussing the current operators it is interesting to terms of this Green’s function. In equilibrium the expecta- consider how the superconducting correlations are modified tion value of the operator can be obtained by evaluating a in the presence of the S/F/S junction. The bulk Green’s sum over the fermionic Matsubara frequencies ωn.Thisstep function of a TI based superconductor [2,3] has mixtures of requires us to extend the Green’s function into the complex even frequency singlet s-wave and triplet p-wave components. plane. We use the fact that the retarded (advanced) Green’s Near the S/F/S interface the translational symmetry is broken function is analytical in the upper (lower) half of the complex and odd-frequency components appear [6,42]. We define the plane. Hence, we perform an analytical continuation from the anomalous spectral function as open interval on the real axis (given by propagating solutions) βE = , , = R − A . to the Matsubara frequencies, by replacing E → iω in all F (x x py E ) (F F ) tanh (18) n 2 expressions in Eq. (15)[56]. To access negative Matsubara R/A frequencies, we calculate the advanced Green’s function in the Here F are off diagonal parts of the corresponding full same manner as the retarded one. The uniqueness of the an- Green’s function obtained in Eq. (15). The superconducting alytical continuation allows us to use these expressions to go GF can be further decomposed into singlet and triplet components to energies below the gap, so the expectation values obtained in this way contain both contributions from the continuum 1 FSE(E ) = dpytr σ0(F (E ) + F (−E )), (19) spectrum and bound states. 2 Finally, the expectation value of the quasiparticle part of the current density operator is given by 3 2 1 ∞ FTO(E ) = dpytr σi(F (E ) − F (−E )) , (20) ev 2 j (x) ≡jˆ (x)= F dp tr[τ σ G (x, x, iω )] i=1 l l 2β y 0 l py n n=−∞ where we have extracted the even- and odd-frequency de- (17) pendence correspondingly. Other parts in the GF have odd with l = x, y [57]. If M = 0, the summation in frequency x momentum dependency and hence vanish after py integration. space for jy(x) does not converge due to an oscillating be- As seen in Figs. 4(a) and 4(b) the spectrum is composed of havior at high energies. This is similar to the behavior of the continuum (E >) and bound states (E <) parts. The jy(x) in the normal state, where the contributions arising from continuum spectrum shows the familiar superconducting peak = the oscillating wave functions for Mx 0 vanish only after at the gap boundary. The spectrum inside the superconduct- integration over x, that is, when computing the transverse ing gap attributed to the bound states is located inside the current from the transverse current density. Such a behavior barrier and decays exponentially away from the barrier. The can also be understood as an artifact of the continuum Dirac even-frequency singlet (SE) part is close to zero inside the model. In fact, this model is only valid close to the Dirac gap and recovers full value deep inside the superconduct- point within the band gap of the TI. To account for this, we ing lead. The odd-frequency (TO) part originating from the separate the current contributions into superconducting and breaking of the translational symmetry in the x direction has = SC + N SC = − N normal parts, j j j , where we define j j j . its maximum value near the S/F interface and decays into N = Here, j is evaluated for a normal system where we set 0 the bulk. The bound states spectrum is equally composed of and captures all divergent terms that we treat in more details SE and TO components. The TO component can be related SC in Appendix E. In the remaining expression j , which is also to Majorana modes at S/F interfaces [59], which in turn the part that does not vanish after integration over x,thesum form Andreev bound states in the junction. We did not find converges fast and is performed numerically up to a cutoff. significant changes in the spectrum of anomalous Green’s Since it can be proven that the normal part goes to zero in function between in-plane and out-of-plane magnetization, so SC equilibrium, we focus only on the regular part j which we cannot attribute the appearance of the odd frequency to the describes the actual Josephson current in the junction. finite transverse supercurrent found in the next section. Note that Eq. (17) only contains the spatial dependence of the quasiparticle part of the current density. In order to VI. JOSEPHSON HALL CURRENT AND CURRENT-PHASE compute the spatial dependence of the full current density, RELATION one also needs to include contributions due to the source term Sˆ(x) from Eq. (12)intheS leads [58]. As a consequence the We are now in a position to discuss the emergence of the full current density in the x direction, consisting of jx (x)from transverse Josephson Hall current, which is the main result of

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FIG. 4. Position dependent spectrum of anomalous Green’s function with magnetization Mx = 0.3 meV. We choose the following param- 5 eters: vF = 5 × 10 m/s, μ = 2meV,d = 330 nm, = 0.2meV,V0 = 0. meV, φ = 0.7π. The broadening of the peaks in the spectrum is η = 0.02. this paper. Without a barrier magnetization, M = 0,orifthere respect to x and consequently the total Josephson Hall current is only an My component of M, the transverse current density through the F region, jy(x) =jˆy(x) vanishes. On the other hand, the asymmetry in d/2 the Andreev spectrum due to a finite Mx or the separation of = , / Iy dxjy(x) (21) Majorana modes localized at the S F interfaces due to a finite −d/2 Mz induce a finite jy(x). This is illustrated by Fig. 5, where we present the spatial dependence of jy(x) in the presence is zero for finite Mz. of a finite magnetization in the barrier. For a magnetization For a magnetization Mx, there is a finite transverse Joseph- Mz [Fig. 5(a)], we observe two transverse current densities of son current density flowing in the same direction inside the opposite sign localized at the S/F interfaces. At each inter- whole F region, as shown in Fig. 5(b). In this case, the face, this localized current density corresponds mainly to the current density profile jy(x)isanevenfunctionofx, which / chiral Majorana mode that emerges at an S F interface for clearly allows for a finite Josephson Hall current Iy as given large Mz as discussed above in Sec. III B. The magnitude of by Eq. (21) flowing in the F region. To increase Iy, one can jy(x) increases proportional to Mz. In contrast to the constant apply an additional gate voltage V0 inside the barrier, which longitudinal Josephson current density, jy(x) oscillates with reduces the effective Fermi momentum in the F region and kF and decays exponentially into the S regions. As shown hence suppresses the oscillating behavior inside the barrier. from a symmetry argument in Appendix F, jy(x) is odd with In Fig. 5(c), one can see that by tuning V0 close to μ we

FIG. 5. Spatial dependence of the transverse current density for different magnetization directions and amplitudes of M and V0:(a)M = Mzez and V0 = 1.5meV,(b)M = Mxex and V0 = 1.5 meV, (c) Mx = 0.1 meV and different V0. In all panels, φ = 0.7π, d = 330 nm, μ = 5 2meV,vF = 5 × 10 m/s, and = 200 μeV.

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VII. CONCLUSIONS In this paper, we have studied Josephson junctions realized on three-dimensional topological insulators which are subject to a Zeeman term in the normal topological insulator region. Most importantly, we have found that the interplay between the spin-momentum locking of the topological insulator surface state, superconductivity, and an in-plane Zeeman field in the normal region gives rise to a net transverse Josephson Hall current. For this Josephson Hall current to emerge, the in-plane Zeeman field has to have a component parallel to the superconducting phase bias direction [see Fig. 1(d)]. Since the effect is caused by an in-plane Zeeman term, we refer to it as FIG. 6. Dependence of averaged Josephson current density the planar Josephson Hall effect to also distinguish it from other Josephson Hall effects [46,47,61]. through the F region (a) on the phase difference φ for Mx = 0.4meV The emergence of the Josephson Hall current is reflected and (b) on the magnetization Mx at φ = 3π/2 for different gating in an asymmetry and ‘tilting’ of the Andreev spectrum with potentials V0. The solid line represents the Josephson Hall current density j¯y and the dotted line represent longitudinal current density respect to the transverse momenta py. If sufficiently large, this 5 jx. In all panels, d = 330 nm, μ = 2meV,vF = 5 × 10 m/s, and asymmetry even induces a transition in the Andreev spectrum = 200 μeV. from a regime with gapless, counterpropagating Majorana modes to a regime with unprotected modes that are unidirectional at small py. Due to strong spin-orbit coupling, the planar Josephson Hall effect in topological-insulator-based junctions can achieve an almost flat profile of j (x) within the junction, y enables sizable Josephson Hall currents, whose amplitudes thereby increasing I . y can be further modulated by electrostatic and/or magnetic For the case of M = 0 we, moreover, compare the x control of the normal region. averaged transverse current density j¯ = I /d with the cor- y y Until now, we have mainly discussed Zeeman terms responding longitudinal Josephson current density j = I /W x x induced into the normal topological insulator region by mag- in Fig. 6 [60]. Here W is the width of the junction, and we netic proximity effects from a nearby ferromagnet, such as in normalized the current to the corresponding junction cross YIG/(Bi, Sb) Te [64], EuS/Bi Se [65], or (Bi,Mn)Te with section. I is proportional to the number of transverse modes, 2 3 2 3 x thin Fe overlayers [66]. Since the planar Josephson Hall ef- i.e., W , and therefore j is effectively W independent. Fig- x fect requires in-plane Zeeman terms, an alternative realization ure 6(a) shows the current-phase relation of j and j¯ for x y could be by applying an in-plane magnetic field along the several different values of V .BothI and I are 2π periodic in 0 x y phase bias direction in the normal region [67]. Assuming, for the superconducting phase difference φ since fermion parity example, an in-plane g factor of g = 10, an in-plane magnetic is not conserved if all states have equilibrium occupations field of around B = 0.35 T corresponds to a Zeeman splitting (see Sec. V). There is, however, a marked difference in the of 0.1 meV [68], which can already yield sizable Josephson current-phase relation between the nonsinusoidal I , which x Hall currents flowing through the normal region, as illustrated is an odd function of φ, and I , which is an even function y by Fig. 6(b). Indeed, in Josephson junctions composed of of φ. Unlike I , I does typically not exhibit zeros at integer x y thin-film aluminium and HgTe quantum wells, which can also multiples of φ = π. Remarkably, we see that for φ close to π act as three-dimensional topological insulators [69], in-plane the direction of the current can be controlled not only by the magnetic fields of more than 1 T have been achieved [70–72]. sign of M but also by modifying the gate voltage V , which x 0 It would be interesting to extend our study to systems can be appealing for practical applications. finite in the y dimension, where the transverse Josephson In Fig. 6(b),weshow j and j¯ at φ = 3π/2 as a function x y current can lead to various phenomena found in sys- of M . This illustrates that for a large enough magnetization x tems with SOC. In this case additional care should be the Josephson Hall current density can exceed the longitudinal taken to account for currents arising in the superconduct- Josephson current. Such ratios j¯ / j > 1 are comparable to y x ing leads, by calculating corrections to the spacial phase the ratios found in normal TI-based ferromagnetic tunneling dependence in a self-consistent manner [73]. Effects such junctions and are a result of the strong SOC in 3D TI surface as current circulation patterns near the edges under hard state. Although in our theoretical treatment we considered wall boundary conditions [73], interplay between longitudi- the limit of the system infinite in the y direction, we believe nal and transverse phase biases in a crossed junction setup that qualitatively our results also hold for the junctions where [74], and control of the phase difference by nonequilib- W and d are comparable. This makes the planar Joseph- rium current injection [75] may occur. Furthermore, here son Hall effect in TI-based Josephson junctions a promising we have focused on transverse charge currents in topolog- candidate to observe sizable transverse currents with ratios ical Josephson junctions. For future research on topological I /I exceeding the corresponding ratios of other Josephson y x Josephson junctions, it might also prove fruitful to study the Hall effects [46,47,61,62]. We also expect the appearance of role of superspin Hall currents [74,76] and spin polariza- transverse Josephson currents (but with smaller I /I ration) y x tions [77–79], known from semiconductor/superconductor or in the Rashba 2DEG ferromagnet junction based on the study ferromagnet/superconductor heterostructures. of the normal planar Hall effect in such a system [63].

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ACKNOWLEDGMENTS cannot be removed even for finite M = 0 (see Refs. [30,31] and Appendix C). The crossing can only be shifted, which O.M. is grateful to the Chair of Theoretical Physics 4 of the is what happens for a finite M = 0, where E (φ) = 0for University of Würzburg for its hospitality. B.S. and E.M.H. y 0 φ = (2n + 1)π − 2Z with n ∈ Z. This protected crossing is acknowledge funding by the Deutsche Forschungsgemein- y a hallmark of the topological Josephson junction and can schaft (DFG, German Research Foundation) through SFB also be found in models with finite F region. At p = 0, the 1170, Project-ID 258499086, through Grant No. HA 5893/4- y main effect of magnetization components M , = 0 is thus to 1 within SPP 1666 and through the Würzburg-Dresden Cluster x z reduce the bandwidth of the ABS and detach them from the of Excellence on Complexity and Topology in Quantum Mat- continuum states. ter – ct.qmat (EXC 2147, Project-ID 390858490) as well as The finite magnetization M = 0 acts as a vector potential by the ENB Graduate School on Topological Insulators. y in the x direction and can be absorbed in the phase difference by performing the gauge transformation of the BdG Hamilto- δ APPENDIX A: ANDREEV BOUND STATES IN THE nian Ux = exp [iτzχ(x)] where MODEL ⎧ − / , < − / , ⎨⎪ Zy 2 x d 2 1. Ansatz and boundary conditions χ(x) = Myx/vF , |x| < d/2, (A4) In Sec. III of the main text, we have presented ABS ob- ⎩⎪ / , > / . tained numerically for a finite F region. Most of the prominent Zy 2 x d 2 features of short TI-based Josephson junctions are, however, So, the gauge invariant phase difference is given by φ − 2Z , δ y already captured by the model of a -like F region with which explains the shift of ABS found in Eq. (A3). h(x) = dδ(x), (x) = , and (x) = φ (x)inEqs.(1) and We also remark that the case of py = 0 is equivalent to (2). The advantage of this model is that it allows for a trans- a Josephson junction based on a single quantum spin Hall parent analytical treatment of the ABS with relatively compact edge if Mx → My, My →−Mz and Mz → Mx. With these expressions. replacements, Eq. (A3) describes the ABS spectrum of such δ For a junction, the ansatz to obtain the ABS is similar Josephson junctions in the short junction regime [82]. to Eq. (7), with the states ψ(x < 0) given by the first line of Eq. (7) and ψ(x > 0) given by the third line of Eq. (7). Now, 3. δ model in Andreev approximation the coefficients A1, A2, B1, and B2 have to be calculated from the boundary conditions at x = 0. This boundary condition Another limit that allows for closed analytical solutions is can be obtained by integrating Eq. (3) from x =−η to x = η the case of |μ|, where we can make use of the Andreev with η → 0+. The corresponding procedure [16,38,80] yields approximation. If we introduce the angle −π/2 <θ<π/2 via vF py = μ sin θ, the eigenstates (4) are simplified within Uˆ+ 0 ψ(0+) = ψ(0−), (A1) the Andreev approximation in so far that 0 Uˆ− √ 2 − E 2 sμv q± ≈ μ θ ± , where F cos i θ cos Z sin Z sin Z ∓ x ∓ − vF (αqξ + ipy ) αθ ∓iZ cos Z Z i Z ( Zz Z0 ) ≈ α i . Uˆ± = e y (A2) μ + ξ e (A5) sin Z ± − ± Zx sin Z ( ) i Z ( Zz Z0 ) cos Z Z With these approximations, the condition for a nontrivial with Z = V d/v , Z = M d/v with l = x, y, z, and Z = 0 0 F l l F solution to Eq. (3) can be written as 2 − 2 − 2 Z0 Zx Zz . X 2 − 2A(θ )X − B(θ ) = 0, (A6) √ 2 2 2. δ model at py = 0 where X = − E /E and = α + First, we look at the case of py 0, where vF ( pξ Zx sin Z cos Z sin θ cos θ / μ + ξ = α A(θ ) = , ipy ) ( ) . We invoke the boundary condition (A1) φ (Z2−Z2 ) sin2 Z 2 θ 2 + + 2 θ 0 x on the first and third lines of Eq. (7) and require a nontrivial Z cos cos 2 Zy sin Z2 solution of the resulting system of linear equations. This then φ 2+ 2 2 2 θ 2 + Z + (Zx Zz ) sin Z yields the two ABS energies E = PE0(φ), where P =±1 cos sin 2 y Z2 B(θ ) = . (A7) denotes the two fermion-parity branches and φ (Z2−Z2 ) sin2 Z 2 θ 2 + + 2 θ 0 x cos cos Zy sin 2 φ/ + 2 Z φ = cos ( 2 Zy ) . E0( ) (A3) From the two solutions of Eq. (A6), X = A(θ ) ± 2 + 2 2 / 2 cos Z Z0 sin Z Z A2(θ ) + B(θ ), one can see that at a fixed angle θ the two solutions for the energy E±(φ,θ) do not come as The two ABS given by E =±E0(φ) exhibit a nondegenerate E±(φ,θ) =−E∓(φ,θ)ifA(θ ) = 0. This is the case for finite zero-energy crossing at φ = π if M = 0 [81]. At this crossing, Zx and finite θ. Instead, the two solutions can be obtained as the ground-state fermion parity changes, and the two branches in Eq. (A3) have been chosen such that each branch pre- sgn A(θ ) ± A2(θ ) + B(θ ) E±(φ,θ) = , (A8) serves its fermion parity [30,31]. As such a nondegenerate 2 zero-energy crossing is protected by the fermion parity, it 1 + A(θ ) ± A2(θ ) + B(θ )

054508-9 OLEKSII MAISTRENKO et al. PHYSICAL REVIEW B 103, 054508 (2021) which only satisfies the weaker condition E±(φ,θ) = ABS in the vicinity of the protected crossing at φ = π − 2Zy −E∓(φ,−θ ) originating from the particle-hole symmetry of for small py. the formalism. While such asymmetric ABS could also be To do so, we first note that the BdG Hamiltonian (2) can be obtained within a semiclassical treatment taking into account written as HˆBdG(py ) = HˆBdG(py = 0) + vF pyσyτz, where we only phase effects, such a treatment does not capture the exact treat the term vF pyσyτz as a perturbation. Then, we can take details of the Andreev spectrum as obtained from a micro- the two parity-conserving ABS |± for py = 0 discussed in scopic treatment such as the one provided here. Sec. A2 and project the full Hamiltonian (2) onto these two If θ = 0, Eq. (A8) reduces simply to E±(φ,θ = 0) = states. This procedure yields the effective Hamiltonian ±|E (φ)| with E (φ) given by Eq. (A3). Note that now the 0 0 ˆ = φ σ + σ + σ , sign ± does not refer to the parity branch, but instead to pos- Heff E0( )˜z v0 py ˜y vy py ˜0 (B1) itive and negative energies. Another point worth mentioning where E (φ) is given by Eq. (A3) and originates from = 0 with regard to Eq. (A8) is that for Zx 0 it reduces to [3,83] HˆBdG(py = 0). In Eq. (B1), the two-level system formed by = thetwoABSatpy 0 is described by the Pauli matrices φ 2 2 σ˜ (l = x, y, z) and the corresponding 2 × 2 unit matrixσ ˜ . 2 Zz sin Z l 0 E±(φ,θ) =± 1 − T (θ ) sin + Z + , y 2 Moreover, we have introduced the velocities 2 Z (A9) Z2 sin2 Z Z0 sin Z + z cos Z + μ 1 2 where v = Z Z v (B2) 0 2 + μ2 Z2+Z2 F 1 + x z sin2 Z cos2 θ Z2 T (θ ) = (A10) Z2 sin2 θ+Z2 cos2 θ sin2 Z 2 θ + ( 0 z ) and cos Z2 Z0 sin Z − μ cos Z Zx sin Z v = Z Z v , (B3) is the transmission of a normal/ferromagnet/normal junction y 2 2 2+ 2 F + μ + Zx Zz 2 1 2 sin Z with Zx = 0[38]. Z = For Mx 0, Eq. (A8) exhibits several salient features: A which arise from the matrix elements of the perturbation M = finite x 0 introduces not only an asymmetry in the ABS vF pyσyτz. Since we are mainly interested in the ABS close p spectrum at finite y, but can even lead to a situation where to the crossing at φ = π − 2Zy, we have approximated the y v ∝ ∂E± φ,θ /∂θ the group velocities in the direction, g ( ) , φ-dependent velocities v0(φ) and vy(φ)byφ-independent p = have the same sign for ABS in the vicinity of y 0 and velocities v0(φ) ≈ v0(π − 2Zy ) ≡ v0 and vy(φ) ≈ vy(π − φ ≈ π − Z 2 y. At these momenta, the two ABS propagate 2Zy ) ≡ vy. in the same direction. This change from nonchiral, counter- ± φ = ± The spectrum of Eq. (B1)isgivenbyEeff ( ) vy py propagating ABS to unidirectional ABS propagating in the 2 2 E (φ) + (v0 py ) . At the crossing point, E0(π − 2Zy ) = same direction occurs for B(θ ) < 0. Close to φ ≈ π − 2Zy, 0 θ < | | > | | ± π − = ± = B( ) 0 is satisfied if Mx V0 . Hence, if the Zeeman term 0 and Eeff ( 2Zy ) (vy v0 )py.Ifvy 0, that is, if φ = φ = π − ± π − in the direction of the phase bias exceeds the mismatch Mx 0, the spectrum at 2Zy is simply Eeff ( between the chemical potentials of the S and F regions, the 2Zy ) =±v0 py and describes two counterpropagating Majo- ABS close to py = 0 and φ + 2Zy ≈ π propagate in the same rana modes along the y direction, similar to Ref. [1]. For finite direction in short junctions. vy, on the other hand, the group velocities (vy ± v0 )ofthetwo At this point, it is important to remark that the Andreev modes point into the same direction if |vy| > |v0|. approximation (A5) breaks down at large transverse momenta, The appearance of a term vy pyσ˜0 in Eq. (B1) has thus its that is, at momenta close to the Fermi momentum pF . Because origin in the unidirectional modes at small py. While there of this, Eq. (A8) does not describe the ABS for py ≈ pF well. is always a finite v0 in TI-based Josephson junctions, vy = 0 This is also illustrated by Fig. 2(d), which shows a comparison only arises for finite Mx = 0. This can be understood in the between Eq. (A8) and the results for a finite barrier without following way: The terms containing v0 and vy originate from any further approximations. For small py = μ sin θ,Eq.(A8) the matrix elements vF pyP|σyτz|P with P, P =±1 denot- is in good agreement with the results of the finite barrier. ing the two parity branches of py = 0. If Mx = 0, the effective Equation (A8) cannot, however, capture the appearance of spin orientation of the eigenstates |± of HˆBdG(py = 0) lie in zero-energy ABS that occur at large momenta once the modes the xz plane and thus the expectation values ±|σyτz|± vanish = close to py 0 become unidirectional. and vy = 0. Only off-diagonal matrix elements ∓|σyτz|± are finite and give rise to v0 = 0. For finite Mx = 0, however, the effective spin expectation APPENDIX B: EFFECTIVE LOW-ENERGY MODEL values of |± now also acquire a component in the y direction The asymmetry of the ABS spectrum as well as the emer- and ±|σyτz|± = 0. The eigenstates |± satisfy the relation gence of unidirectional modes for large Mx and small py can |± = Kˆ |∓, where Kˆ denotes complex conjugation. Because be understood from the interplay between the effective spin of this property, +|σyτz|+ = −|σyτz|− and consequently degree of freedom and Mx. To elucidate the origin of these the diagonal matrix elements of the perturbation is propor- modes, we employ a simple effective low-energy Hamilto- tional toσ ˜0 (and not toσ ˜z or a linear combination ofσ ˜0 andσ ˜z). nian. For a δ-like F region, the BdG Hamiltonian (2) always The spectrum of Eq. (B1) makes it clear that the ABS spec- supports two ABS. Following the procedure in Ref. [1], we de- trum for small py and close to the protected crossing point φ + rive an effective low-energy Hamiltonian describing these two 2Zy = π (or, more generally, close to φ = (2n + 1)π − 2Zy

054508-10 PLANAR JOSEPHSON HALL EFFECT IN TOPOLOGICAL … PHYSICAL REVIEW B 103, 054508 (2021) with n ∈ Z) can support unidirectional modes around py ≈ 0 We remind the reader that because of the rotated spin axes for finite Mx. used in Eq. (2) M in Eq. (C6) is a rotated effective magnetization. This magnetization M is related to the components = , , APPENDIX C: PROTECTED ZERO-ENERGY CROSSING of the real magnetization M (Mx My Mz ) induced in the F = − , , FOR py = 0 region via M ( My Mx Mz ). In Eq. (C5), the additional Cˆ {Cˆ, ˆ }= terms behave differently under : HM 0 and thus a A peculiar feature of the ABS spectrum of a Josephson finite M does not remove the zero-energy crossing. On the junction based on a single surface of a 3D TI is its protected other hand, {Cˆ, Hˆ } = 0 and thus a gap is opened at finite py zero-energy crossing for p = 0, even in the presence of a py y because in this case particle-hole symmetry does not protect Zeeman term in the F region, as discussed in Sec. A2. Follow- Hˆ but connects Hˆ and Hˆ (see above). While we have py py −py ing Ref. [31], we can understand this protection arising from δ the particle-hole symmetry of the BdG Hamiltonian, which employed this analysis to the case of a barrier for illustration, we note that this is valid for all single-energy crossings that are allows one to define a Pfaffian, Pf[HˆBdG(py = 0)] for any φ. The existence of a Pfaffian then implies that twofold de- only double degenerate, including the case of a finite barrier generate zero-energy states are generically protected against also studied in this paper [84]. perturbations as long as particle-hole symmetry is preserved. Hence, as a final remark we note that the analysis from Eqs. (C3) and (C4) applies also to the case of finite M, For the system studied here, particle-hole symmetry is de- φ/ φ scribed by the operator Cˆ = σ τ Kˆ , where Kˆ denotes complex where cos( 2) should simply be replaced by E0( ) y y from Eq. (A3). This also makes it clear that the ground- conjugation and σy and τy are the respective Pauli matrices in − F0 = φ = / state fermion parity F0 given by ( 1) sgn[E0( )] spin and electron hole space. Any BdG Hamiltonian, includ- φ/ + φ ing Eq. (2), anticommutes with Cˆ, sgn[cos( 2 Zy )] for finite M is only shifted in its dependence by Zy ∝ My, but remains unaltered otherwise. {Cˆ, HˆBdG}=0. (C1)

If we introduce the momentum quantum number py, this be- APPENDIX D: ADDITIONAL ANDREEV BOUND comes STATES RESULTS Cˆ ˆ Cˆ−1 =−ˆ − . HBdG(py ) HBdG( py ) (C2) 1. Large out-of-plane Zeeman term = Thus, only for py 0, does particle-hole symmetry imply In this part, we provide additional plots of the ABS and {Cˆ, HˆBdG(py = 0)}=0, while in general particle-hole sym- discuss the relevant system parameters required to observe the metry connects states with py to states with −py. effect of detaching the Andreev bound states from the contin- From now on, we focus only on the two ABS |± at uum spectrum. Inducing a bigger gap between the Andreev py = 0 and with M = 0.Foraδ-like F region and M = 0, bound states and the continuum states requires increasing the =± φ/ the energies are simply given by E cos( 2), that is, effective barrier strength Zz. This can be achieved either by a φ = π they possess twofold degenerate zero-energy states at . stronger magnetic field or magnetization Mz or by increasing Similar to Sec. B, the corresponding low-energy Hamiltonian the barrier length d. Here we keep d relatively small, i.e., is the 2 × 2 matrix with respect to the ABS |±, in the short-junction limit, in order to be comparable with δ cos(φ/2) 0 our analytical -barrier solution. On the other hand, large Mz Hˆ 0 = , (C3) requires large g factors and huge magnetic fields which are eff − φ/ 0 cos( 2) not feasible in experiments. If the Zeeman term is proximity which can in turn be transformed to induced from a nearby ferromagnet on the other hand larger φ/ Mz are indeed possible. Figure 7 shows the results for such a 0 0 cos( 2) 0 H˜ˆ = i ≡ iAˆ . (C4) case, where we increased the magnetization to Mz = 1meV. eff − cos(φ/2) 0 eff ˆ 0 = The Pfaffian of Eq. (C3) is then given by Pf(Heff ) 2. Chemical potential dependence ˆ0 = φ/ iPf(Aeff ) i cos( 2) and can be related to the ground-state − F0 = ˆ0 ˆ 0 Although more difficult to control experimentally, our fermion parity F0 via ( 1) sgn[Pf(Aeff )]. Since Pf(Heff ) exhibits only a single zero, a perturbation that preserves model also allows us to consider a situation where the chem- μ particle-hole symmetry cannot remove the two zero-energy ical potential in the proximitized S leads is similar to the μ ∼ states, but only shift them to other values of φ [31]. induced superconducting gap, i.e., . This situation is Now, we are in a position to understand why the crossing at beyond the Andreev approximation commonly used in previ- φ = π is protected against finite M in the F region. For finite ous studies. We show the Andreev bound states for this regime in Fig. 8. Similar to the case of large μ, a finite Mx leads to M and py, we can write the BdG Hamiltonian as an asymmetry in the Andreev spectrum with respect to the Hˆ (p ) = Hˆ (p = 0)| = + Hˆ + Hˆ (C5) BdG y BdG y M 0 M py transverse momentum ky, which in turn gives rise to a transverse Josephson Hall current. A main effect of the decreased with chemical potential is that larger values of Mx are required to ˆ =− · σ = HM M h(x)(C6)achieve the regime with unidirectional modes close to ky 0. This can be seen by comparing parameters used in Fig. 8(b) and and the ones in Fig. 2(d) in the main text. Figure 8 also Hˆ = v p σ τ . (C7) py F y y z illustrates that for small chemical potentials Andreev bound

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To obtain the Green’s function, we proceed analogously to the main text. For example, the lead solutions are 1 (0) 1 iαq x ψ (x) = √ αq0 + ipy e 0 , (E2) α v 2 F μ + E 2 2 where vF q0 = (μ + E ) − (vF py ) and α =±1givesthe direction of propagation. In this case, we have two helical counterpropagating sates for each py. The F-barrier solution is given by Eq. (6) with ξ =+1. We omit the details of solving the transposed Hamiltonian and deriving the scattering states. To analyze the current expectation value, we stay within FIG. 7. Dependence of the Andreev bound states on (a) phase the real-energy picture because it allows for a discussion of and (b) transverse momentum. Here, we used the parameters high-energy contributions and the continuation to the Mat- 5 vF = 5 × 10 m/s, μ = 2meV,d = 330 nm, = 0.1meV,V0 = subara frequencies when the function does not decay fast for | |→∞ N = σ 0.2meV,andMz = 1 meV. Solid lines are analytical expressions for E . The current operator simplifies to j evF and δ the -barrier model and dots are obtained from the numerical solution we obtain of a finite-barrier problem. jN (x) =−2ev dE n(E ) dp trσ GR (x, x) , (E3) i F y i py states appear at momenta ky larger than kF , which is very where n(E ) is the Fermi-Dirac distribution. In the case different from the commonly studied case of μ . |E + μ| |py|, we can conduct a variable substitution in the integral, namely vF q0 =|E + μ| cos θ and vF py = APPENDIX E: NORMAL JUNCTION GREEN’S FUNCTION |E + μ| sin θ. This allows us to rewrite the integration limits over py, which yields for the transverse current Let us consider the simpler case of a normal junction to bet- ter understand the asymptotic behavior of the superconducting jN (x < −d/2) = 2ev dE|E + μ|n(E ) solution for |E|||. If we switch off superconductivity y F = by putting 0, there must be no current in equilibrium. π/2 − μ+ θ/ − αθ However, calculating the expectation value of the transverse dθ r(E,θ)e 2ix( E )cos vF e i , current has some technical difficulties which we address in −π/2 this section. Without the superconductor, the Hamiltonian is (E4) defined by where r(E,θ) is the reflection coefficient of the mode inci- | + μ| < | | H = v σ · pˆ − μ + (V − M · σ)h(x). (E1) dent from the left lead. The case of E py is treated N F 0 analogously employing hyperbolic functions. In the nonsuperconducting case, it is possible to obtain a relatively compact form for the reflection coefficient [38]. δ-barrier solutions.. We consider the stationary states similar to Eq. (13), with the superconducting lead wave functions ψ (0) → replaced by n (x) and in the limit d 0. Then, using the boundary condition (A1) for the electron block, we obtain the reflection and transmission coefficients eisθ (Z + iZ cos θ − sZ sin θ )sinZ r = x z 0 , (E5) Z cos θ cos Z + i(Z0 − sZx sin θ )sinZ − e iZy Z cos θ t = , (E6) Z cos θ cos Z + i(Z0 − sZx sin θ )sinZ where we have defined s = sgn(E + μ). We notice the property r(E,θ) = r(−E, −θ ). The δ barrier does not introduce an energy scale. Therefore, the reflection amplitude is energy independent. This means that all states in the system are affected by the introduction of the barrier, which has significant consequences for Eq. (E4) because the spectrum is not bounded from below. Finite-barrier solutions.. By using the boundary conditions from Eq. (8) and lead wave functions defined in Eq. (E2), FIG. 8. Dependence of the Andreev bound states on the trans- we obtain the reflection and transmission coefficients as verse momentum for μ = 0.2 meV and different Mx at (a) φ = 0.5π θ− +μ θ/ φ = π = × 5 / is id(E )cos vF and (b) . Here, we use the parameters vF 5 10 m s, = e a(E )sin(dk0 ) , d = 330 nm, = 0.1meV,M = 0, M = 0, and V = 0. r(E ) (E7) y z 0 vF k0 cos θ cos(dk0 ) + ib(E )sin(dk0 )

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the junction Hamiltonian (2) have the following properties: − / − +μ θ/ idMy vF id(E )cos vF θ = e vF k0 cos (x) = (−x), h(x) = h(−x), and we can choose (x) = t(E ) (E8) − − vF k0 cos θ cos(dk0 ) + ib(E )sin(dk0 ) ( x) because only the relative phase is important. Appli- T = σ K with s = sgn(E + μ) and cation of the time-reversal symmetry i y results in the following changes in the Hamiltonian: →− and M → − →− I a(E ) = Mx + iMz cos θ − sV0 sin θ, (E9) M (equivalent to M M). Inversion symmetry has the 2 effect of →− and pˆ →−pˆ, but does not change the spin. b(E ) = (E + μ) cos θ + V0 − Mx sin θ. (E10) We also consider two mirror planes Myz and Mxy, which act In this case, r(E ) exhibits a behavior ∼1/|E| for large |E|,but in the spin space such that My,z →−My,z and Mx,y →−Mx,y, this is still not enough to make the energy integral in Eq. (E4) respectively, and both reverse the sign of the kinetic term. If ﬁnite. Mx(z) = 0, we ﬁnd that Sx(z) = Myz(xy)IT is a symmetry of The divergent behavior comes from the fact that the energy the Hamiltonian. spectrum of the Dirac cone is not bound from below, so for- Next, we derive how the current operator transforms under mally we have to include all contributions down to E =−∞ given symmetries in the expectation values. At the same time, the presence S S−1 =− S S−1 = . of the magnetic barrier introduces spin polarization into all x jy x jy and z jy z jy (F1) states, making them contribute to the integral (E4). In the real Using that Sψ(r) can be presented as Uψ∗(V r), where U is a system, the low-energy model is invalid for energies far from unitary matrix in spinor space and V is an orthogonal trans- the Dirac cone located in the gap of the topological insulator. formation in coordinate space, we obtain a relation for the On the other hand, the high-energy solutions become highly contribution of the operator expectation value from a single / oscillating with wave vector E vF . These oscillations cannot state be resolved in the real system, which provides another argu- Sψ | |Sψ =ψ | S S−1 |ψ , ment why we should drop high-energy terms. Thus, we choose (r) jy (r) (V r) jy (V r) (F2) −λ|E| to use the regularization e in the integral. Then, we can where the scalar product is performed only in spinor space. perform the energy integration analytically which yields a λ The expectation value of the total current density is the sum prefactor in front of the expression for the current. Hence, of contributions from all states weighted with the occupation λ → N after taking the limit 0, jy vanishes. When computing number, which is a function of energy. If S is the symmetry of the current density in the superconducting case (>0), we |ψ , S |ψ , N HBdG, states n(x y) and n(x y) either have the same subtract j expression before performing integration. After energy or coincide. Hence, we obtain that, the integral can be performed numerically or more con- veniently by going to the complex plane and mapping it to the jy(x, y)=−jy(−x, y) if Mx = 0, (F3) Matsubara sum. jy(x, y)=jy(−x, −y) if Mz = 0. (F4) APPENDIX F: SYMMETRIES OF THE CURRENT Since the current is independent of y due to translation invari- OPERATOR ance, these symmetry relations are generalized to the whole We can get some insight into the current operator expec- junction. We also mention that in case of the δ barrier we tation values from a symmetry point of view. In this section, may have a discontinuity at x = 0 and the value of the current we provide the conditions for the current density jy(x) to would depend on the direction from which we approach the be an even or odd function. First, we note that expressions in barrier.

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